In a previous paper by the author joint with Baogang XU published in Discrete Math in 2018, we show that every non-planar toroidal graph can be edge partitioned into a planar graph and an outerplanar graph. This edge ...In a previous paper by the author joint with Baogang XU published in Discrete Math in 2018, we show that every non-planar toroidal graph can be edge partitioned into a planar graph and an outerplanar graph. This edge partition then implies some results in thickness and outerthickness of toroidal graphs. In particular, if each planar graph has outerthickness at most 2(conjectured by Chartrand, Geller and Hedetniemi in 1971 and the confirmation of the conjecture was announced by Gon?calves in 2005), then the outerthickness of toroidal graphs is at most 3 which is the best possible due to K7.In this paper we continue to study the edge partition for projective planar graphs and Klein bottle embeddable graphs. We show that(1) every non-planar but projective planar graph can be edge partitioned into a planar graph and a union of caterpillar trees;and(2) every non-planar Klein bottle embeddable graph can be edge partitioned into a planar graph and a subgraph of two vertex amalgamation of a caterpillar tree with a cycle with pendant edges. As consequences,the thinkness of projective planar graphs and Klein bottle embeddabe graphs are at most 2,which are the best possible, and the outerthickness of these graphs are at most 3.展开更多
文摘In a previous paper by the author joint with Baogang XU published in Discrete Math in 2018, we show that every non-planar toroidal graph can be edge partitioned into a planar graph and an outerplanar graph. This edge partition then implies some results in thickness and outerthickness of toroidal graphs. In particular, if each planar graph has outerthickness at most 2(conjectured by Chartrand, Geller and Hedetniemi in 1971 and the confirmation of the conjecture was announced by Gon?calves in 2005), then the outerthickness of toroidal graphs is at most 3 which is the best possible due to K7.In this paper we continue to study the edge partition for projective planar graphs and Klein bottle embeddable graphs. We show that(1) every non-planar but projective planar graph can be edge partitioned into a planar graph and a union of caterpillar trees;and(2) every non-planar Klein bottle embeddable graph can be edge partitioned into a planar graph and a subgraph of two vertex amalgamation of a caterpillar tree with a cycle with pendant edges. As consequences,the thinkness of projective planar graphs and Klein bottle embeddabe graphs are at most 2,which are the best possible, and the outerthickness of these graphs are at most 3.
